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P-Matrix Reasoning and Information Intelligent Mining

Published Online: 15 Jun 2022
Volume & Issue: AHEAD OF PRINT
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Received: 30 Dec 2021
Accepted: 27 Mar 2022
Journal Details
License
Format
Journal
eISSN
2444-8656
First Published
01 Jan 2016
Publication timeframe
2 times per year
Languages
English
Abstract

P-sets (P stand for Packet) and P-matrix are novel and effective mathematical tools for studying dynamic information systems. In this paper, the concept of P-information mining is given by using the dynamic characteristics of P-sets and P-matrix structure. In addition, the reasoning theorem of P-matrix and the reasoning structure are given. Moreover, the information intelligent mining method under the condition of P-matrix reasoning is obtained. As the application, intelligent recognition of information image was shown.

Keywords

MSC 2010

Introduction

Information (data) mining and applications are the important branch of theoretical and application research in information (data) system and information (data) engineering. Information (data) mining is a dynamic process. For a given source information (x) = {x1,x2 ⋯ ,xq}, ∀xi ∈ (x) is an information element in (x). The concept is given as follows:

Deleting some redundant information elements xi, (x) becomes (x) = {x1,x2 ⋯ ,xp}, p < q; (x) is mined-discovered in (x).

Supplementing some information elements xj in (x), (x) becomes (x)F = {x1,x2 ⋯ ,xr}, q < r; (x)F is mined, i.e. discovered outside (x).

Deleting some redundant information elements xi in (x), and supplementing some information elements xj in (x), (x) becomes (x), and (x) also becomes (x)F, (x) ⊆ (x) ⊆ (x)F; ((x),(x)F) is mined, i.e. discovered inside-outside (x).

I-III are the three different concepts of information (data) mining. A number of authors are studying the dynamic characteristics of I–III to find a mathematical method. In Ref. [1], the concept of P-sets with dynamic characteristics is proposed by improving the finite common element set (Conter set). In Refs. [2,3,4,5,6,7,8,9,10], the application of P-sets in dynamic information system is shown. In Ref. [11], P-matrix (P-augmented matrix) is obtained by using the structure and dynamic characteristics of P-sets. P-matrix reasoning (p-augmented matrix reasoning) is a new reasoning structure with dynamic characteristics generated by P-sets. In this paper, the basic theory and application of information (data) intelligence mining are discussed using the structure and dynamic characteristics of P-sets and P-matrix reasoning.

Conter set X is given, α is the attribute set of X, the dynamic characteristics of P-sets are as follows:

Supplementing some attributes in α, α becomes αF, ααF; X becomes internal P-sets X, XX.

Deleting some attributes in α, α becomes α, αα; X becomes outer P-sets XF, XXF.

Supplementing some attributes in α, and deleting some attributes in α, α becomes αF, and α also becomes α, αααF; X becomes internal P-sets X, and X also becomes outer P-sets XF, XXXF; or X becomes set pair (X,XF). Supplementing attributes be moved into α, and deleting attributes out of α continuously, X becomes some set pairs: (X1F¯,X1F),(X2F¯,X2F),,(XnF¯,XnF) (X_1^{\bar F},X_1^F),(X_2^{\bar F},X_2^F), \ldots ,(X_n^{\bar F},X_n^F) ; (X,XF) is P-sets.

The dynamic characteristics (1–3) of P-sets are the same as those of the information mining concepts I–III. If X,X,XF,(X,XF) are defined as source information (x), internal P-information (x), outer P-information (x)F, P-information ((x),(x)F); or (x) = X, (x) = X, (x)F = XF, ((x),(x)F) = (X,XF), then P-sets give a new mathematical method in the research of information (data) mining.

This paper gives the structures and characteristics of P-sets and P-matrix (P-augmented matrix), gives three new concepts of information mining, proposes P-matrix reasoning (P-augmented matrix reasoning) and reasoning theorems, presents the information (data) intelligent mining method under the condition of P-matrix reasoning and gives the simple application of information (data) intelligent recursive mining. All the results given in this paper are new.

P-sets and its structure

In Ref. [1], P-sets and its structure are presented as follows: Conter set X = {x1,x2,⋯ ,xq} ⊂ U is given, and α = {α1,α2,⋯ ,αk} ⊂ V is the attribute set of X, X is referred to as internal packet set generated by X, referred to as that X is internal packet set for short, XF¯=XX {X^{\bar F}} = X - {X^ - } X is referred to as -element deleted set of X, X={xi|xiX,f¯(xi)=ui¯X,f¯F¯} {X^ - } = \left\{ {{x_i}|{x_i} \in X,\bar f\left( {{x_i}} \right) = {u_i}\bar \in X,\bar f \in \bar F} \right\} If attribute set αF of X meets αF=α{αi'|f(βi)=αi'α,fF} {\alpha ^F} = \alpha \cup \left\{ {\alpha _i^\prime|f({\beta _i}) = \alpha _i^\prime \in \alpha ,f \in F} \right\} Here:

In Eq. (3), βiV, βi¯α {\beta _i}\bar \in \alpha ; fF changes βi into f(βi)=αi'α f({\beta _i}) = \alpha _i^\prime \in \alpha ;

In Eq. (1), X ≠ ∅, X = {x1,x2,⋯ ,xp}, p < q; p,qN+.

Conter set X = {x1,x2,⋯ ,xq} ⊂ U is given, and α = {α1,α2,⋯ ,αk} ⊂ V is the attribute set of X, XF is referred to as outer P-sets generated by X, referred to as that XF is outer packet set for short, XF=XX+ {X^F} = X \cup {X^ + } X+ is referred to as F-element supplemented set of X, X+={ui|uiU,ui¯X,f(ui)=xi'X,fF} {X^ + } = \{ {u_i}|{u_i} \in U,{u_i}\bar \in X,f({u_i}) = {x_i}^\prime \in X,f \in F\} If attribute set α of XF meets αF¯=α{βi|f¯(αi)=βi¯α,f¯F¯} {\alpha ^{\overline F }} = \alpha - \left\{ {{\beta _i}|\bar f({\alpha _i}) = {\beta _i}\bar \in \alpha ,\bar f \in \bar F} \right\} Here:

In Eq. (6), αiα, changes αi into f¯(αi)=βi¯α \bar f({\alpha _i}) = {\beta _i}\bar \in \alpha , α ≠ ∅;

In Eq. (4), XF = {x1,x2,...,xr},q < r;q,rN+.

The set pair which is composed of internal packet set X and outer packet set XF is referred to as P-sets generated by X, referred to as P-sets for short and written as (XF¯,XF) ({X^{\bar F}},{X^F}) Conter set X is referred to as the ground set of P-sets.

From Eq. (3), α1Fα2Fαn1FαnF \alpha _1^F \subseteq \alpha _2^F \subseteq \cdots \subseteq \alpha _{n - 1}^F \subseteq \alpha _n^F According to Eq. (8), we get internal P-sets chain XnF¯Xn1F¯X2F¯X1F¯ X_n^{\bar F} \subseteq X_{n - 1}^{\bar F} \subseteq \cdots \subseteq X_2^{\bar F} \subseteq X_1^{\bar F} From Eq. (6), αnF¯αn1F¯α2F¯α1F¯, \alpha _n^{\bar F} \subseteq \alpha _{n - 1}^{\bar F} \subseteq \cdots \subseteq \alpha _2^{\bar F} \subseteq \alpha _1^{\bar F}, According to Eq. (10), we get outer P-sets chain X1FX2FXn1FXnF X_1^F \subseteq X_2^F \subseteq \cdots \subseteq X_{n - 1}^F \subseteq X_n^F By using Eqs (9) and (11), we get {(XiF¯,XjF)|iI,jJ} \left\{ {(X_i^{\bar F},X_j^F)|i \in I,j \in J} \right\} Eq. (12) is referred to as the family of P-sets generated by X; and Eq. (12) is also the general expression of P-sets.

From Eqs (1)–(7) and (12), we get the following proposition.

Proposition 1

Under the condition F = = ∅, P-sets (X, XF) and Conter set X meet: (XF¯,XF)F=F¯==X. {({X^{\bar F}},{X^F})_{F = \bar F = \emptyset }} = X.

Proposition 2

Under the condition F = = ∅, the family of P-sets {(XiF¯,XjF)|iI,jJ} \{ (X_i^{\bar F},X_j^F)|i \in I,j \in J\} and Conter set X meet: {(XiF¯,XjF)|iI,jJ}F=F¯==X. {\left\{ {(X_i^{\bar F},X_j^F)|i \in I,j \in J} \right\}_{F = \bar F = \emptyset }} = X.

Remark

U is a finite element domain and V is a finite attribute domain.

fF and are element (attribute) transfer; F = { f1, f2,⋯ , fn} and = { 1, 2,⋯ , n} are the family of element (attribute) transfer. The characteristic of fF are that: for element uiU, ui¯X {u_i}\bar \in X , fF changes ui into f(ui)=xi'X f({u_i}) = x_i^\prime \in X ; for attribute βiV, βi¯α {\beta _i}\bar \in \alpha , fF changes βi into f (βi) = αiα. The characteristic of are that: for element xiX, changes xi into f¯(xi)=ui¯X \bar f({x_i}) = {u_i}\bar \in X . For attribute αi ∈ α, changes αi into f¯(αi)=βi¯α \bar f({\alpha _i}) = {\beta _i}\bar \in \alpha . Element (attribute) transfer is a concept of function, transformation.

The dynamic characteristic of Eq. (1) are the same as the dynamic characteristics of down-counter T = T − 1.

The dynamic characteristic of Eq. (4) are the same as the dynamic characteristics of accumulator T = T + 1. For example, in Eq. (4), X1F = XX1+; let X1F = X, we get X2F = X1FX2+ = (XX1+) ∪ X2+,⋯, and so on.

Fact and evidence of existence of P-sets

Take an example: X = {x1,x2,x3,x4,x5} is a finite commodity element set of five apples, and α = {α1,α2,α3} is the attribute set confined in X, where α1 denotes red colour, α2 denotes sweet taste, α3 denotes produced in Henan province of China. Obviously, ∀xiX,i = 1,2,⋯, 5, xi has attributes α1,α2 and α3; the attribute αi of xiX meets ‘conjunctive normal form’; moreover αi=α1α2α3=t=13αt. {\alpha _i} = {\alpha _1} \wedge {\alpha _2} \wedge {\alpha _3} = \wedge _{t = 1}^3{\alpha _t}. Let α4 denotes weight is 150 g, supplementing attribute α4 in α, α is changed into αF = α ∪ {α4} = {α1,α2,α3, α4}, and X is changed into internal P-sets X = {x1,x2,x5}. Obviously, ∀xiX, i = 1,2,5, xi has attributes α1,α2,α3 and α4. The attribute αi of xi meets ‘conjunctive normal form’ expansion; moreover αi=(α1α2α3)α4=(t=13αt)α4=t=14αt. {\alpha _i} = \left( {{\alpha _1} \wedge {\alpha _2} \wedge {\alpha _3}} \right) \wedge {\alpha _4} = \left( { \wedge _{t = 1}^3{\alpha _t}} \right) \wedge {\alpha _4} = \wedge _{t = 1}^4{\alpha _t}. If deleting attribute α3 in α, α is changed into α = α − {α3} = {α1,α2}, and X is changed into outer P-sets XF = {α1,α2,α3,α4,α5,α6,α7}. Obviously, ∀xiXF, i = 1,2,3,4,5,6,7, xi has attributes α1,α2. The attribute αi of xi meets ‘conjunctive normal form’ contraction; moreover αi=(α1α2α3)α3=(t=13αt)α3=t=12αt. {\alpha _i} = \left( {{\alpha _1} \wedge {\alpha _2} \wedge {\alpha _3}} \right) - \wedge {\alpha _3} = \left( { \wedge _{t = 1}^3{\alpha _t}} \right) - \wedge {\alpha _3} = \wedge _{t = 1}^2{\alpha _t}.

The structure and generation of P-augmented matrix

In Ref. [11], the structure and generation of p-augmented matrix are proposed as follows: Conter set X = {x1,x2,⋯ ,xq} is given, xiX has element values y1,i,y2,i,⋯ ,ym,i; yi = (y1,i,y2,i,⋯ ,ym,i)T is a vector generated by element values y1,i,y2,i,⋯ ,ym,i of xi, i = 1,2,⋯ ,q; refer to A=(y1,1y1,2y1,qy2,1y2,2y2,qym,1ym,2ym,q) A = \left( {\matrix{ {{y_{1,1}}} & {{y_{1,2}}} & \cdots & {{y_{1,q}}} \cr {{y_{2,1}}} & {{y_{2,2}}} & \cdots & {{y_{2,q}}} \cr \vdots & \vdots & \ddots & \vdots \cr {{y_{m,1}}} & {{y_{m,2}}} & \cdots & {{y_{m,q}}} \cr } } \right) as matrix of element values generated by X.

Internal P-sets X = {x1,x2,⋯ ,xp} is given, xiX has element values y1,i,y2,i,⋯ ,ym,i; yi = (y1,i,y2,i,⋯, ym,i)T is a vector generated of y1,i,y2,i,⋯ ,ym,i, i = 1,2,⋯ , p; AF¯=(y1,1y1,2y1,py2,1y2,2y2,pym,1ym,2ym,p) {A^{\bar F}} = \left( {\matrix{ {{y_{1,1}}} & {{y_{1,2}}} & \cdots & {{y_{1,p}}} \cr {{y_{2,1}}} & {{y_{2,2}}} & \cdots & {{y_{2,p}}} \cr \vdots & \vdots & \ddots & \vdots \cr {{y_{m,1}}} & {{y_{m,2}}} & \cdots & {{y_{m,p}}} \cr } } \right) is generated by X, and is called internal P-augmented matrix of A.

Outer P-sets XF = {x1,x2,⋯ ,xr} is given, xiXF has element values y1,i,y2,i,⋯ ,ym,i; yi = (y1,i,y2,i,⋯, ym,i)T is a vector generated of y1,i,y2,i,⋯ ,ym,i, i = 1,2,⋯ ,r; AF=(y1,1y1,2y1,ry2,1y2,2y2,rym,1ym,2ym,r) {A^F} = \left( {\matrix{ {{y_{1,1}}} & {{y_{1,2}}} & \cdots & {{y_{1,r}}} \cr {{y_{2,1}}} & {{y_{2,2}}} & \cdots & {{y_{2,r}}} \cr \vdots & \vdots & \ddots & \vdots \cr {{y_{m,1}}} & {{y_{m,2}}} & \cdots & {{y_{m,r}}} \cr } } \right) is generated by XF, and is called outer P-augmented matrix of A.

Here: in Eqs (15)–(17), pqr; p,q,rN+.

The matrix pair composed of A and AF is called P-augmented matrix generated by (X,XF), or P-augmented matrix (AF¯,AF) ({A^{\bar F}},{A^F}) The matrix family {(AiF¯,AjF)|iI,jJ} \left\{ {(A_i^{\bar F},A_j^F)|i \in I,j \in J} \right\} generated by {(XiF¯,XjF)|iI,jJ} \left\{ {(X_i^{\bar F},X_j^F)|i \in I,j \in J} \right\} is called P-augmented matrix family; Eq. (19) is the common form of P-augmented matrix.

It should be noted that: In Ref. [11], by using the structure and dynamic characteristic of P-sets, they improved the definition of augmented matrix in classical mathematics, proposed internal P-augmented matrix A, outer P-augmented matrix AF, and P-augmented matrix (A,AF). The concept of outer P-augmented matrix AF is similar to the concept of augmented matrix in classical mathematics. The definitions of internal P-augmented matrix A and P-augmented matrix (A,AF) cannot be found in classical mathematics. In Ref. [11], under the certain conditions they proved A,AF and (A,AF) return to augmented matrix in classical mathematics. The preliminary concepts in Sections 2 and 3 are important for the research and results of Sections 4 and 5 in thesis. These concepts are used in Sections 4 and 5.

Assumption

The internal P-augmented matrix A, outer P-augmented matrix AF, and P-augmented matrix (A,AF) in Section 3 are called internal P-matrix A, outer P-matrix AF, and P-matrix (A,AF) for short in the following discussions. X,X,XF,(X,XF) are denoted by (x),(x),(x)F and ((x),(x)F), or (x) = X,(x) = X,(x)F = XF, ((x),(x)F) = (X,XF); (x), (x),(x)F and ((x),(x)F) are called information, internal P-information, outer P-information and P-information; ∀xi ∈ (x)(or ∀xi ∈ (x), or ∀xi ∈ (x)F) is called information element, its numerical value yi is called information value of xi. The matrix A in Eq. (15) is called information value matrix.

P-matrix reasoning structure and reasoning theorem

If internal P-matrix Ak+1F¯ A_{k + 1}^{\bar F} , AkF¯ A_k^{\bar F} and internal P-information (x)k+1F¯ (x)_{k + 1}^{\bar F} , (x)kF¯ (x)_k^{\bar F} meet ifAk+1F¯AkF¯,then(x)k+1F¯(x)kF¯. {\rm{if}}\;A_{k + 1}^{\bar F} \Rightarrow A_k^{\bar F},\;{\rm{then}}\;(x)_{k + 1}^{\bar F} \Rightarrow (x)_k^{\bar F}. Eq. (20) is called internal P-matrix reasoning generated by internal P-matrix; Ak+1F¯AkF¯ A_{k + 1}^{\bar F} \Rightarrow A_k^{\bar F} is called the condition of internal P-matrix reasoning, (x)k+1F¯(x)kF¯ (x)_{k + 1}^{\bar F} \Rightarrow (x)_k^{\bar F} is called the conclusion of internal P-matrix reasoning. In Eq. (20), Ak+1F¯AkF¯ A_{k + 1}^{\overline F } \Rightarrow A_k^{\overline F } is equal to Ak+1F¯AkF¯ A_{k + 1}^{\overline F } \subseteq A_k^{\overline F } , (x)k+1F¯(x)kF¯ (x)_{k + 1}^{\bar F} \Rightarrow (x)_k^{\bar F} is equal to (x)k+1F¯(x)kF¯ (x)_{k + 1}^{\bar F} \subseteq (x)_k^{\bar F} .

If outer P-matrix Ak+1F A_{k + 1}^F , AkF A_k^F and outer P-information (x)k+1F (x)_{k + 1}^F , (x)kF (x)_k^F meet ifAkFAk+1F,then(x)kF(x)k+1F. {\rm{if}}\;A_k^F \Rightarrow A_{k + 1}^F,\;{\rm{then}}\;(x)_k^F \Rightarrow (x)_{k + 1}^F. Eq. (21) is called outer P-matrix reasoning generated by outer P-matrix; AkFAk+1F A_k^F \Rightarrow A_{k + 1}^F is called the condition of outer P-matrix reasoning, (x)kF(x)k+1F (x)_k^F \Rightarrow (x)_{k + 1}^F is called the conclusion of outer P-matrix reasoning.

If P-matrix (Ak+1F¯,AkF) (A_{k + 1}^{\bar F},A_k^F) , (AkF¯,Ak+1F) (A_k^{\bar F},A_{k + 1}^F) and P-information ((x)k+1F¯,(x)kF) ((x)_{k + 1}^{\bar F},(x)_k^F) , ((x)kF¯,(x)k+1F) ((x)_k^{\bar F},(x)_{k + 1}^F) meet: if(Ak+1F¯,AkF)(AkF¯,Ak+1F),then((x)k+1F¯,(x)kF)((x)kF¯,(x)k+1F). {\rm{if}}\;(A_{k + 1}^{\bar F},A_k^F) \Rightarrow (A_k^{\bar F},A_{k + 1}^F),\;{\rm{then}}\;((x)_{k + 1}^{\bar F},(x)_k^F) \Rightarrow ((x)_k^{\bar F},(x)_{k + 1}^F). Eq. (22) is called P-matrix reasoning generated by P-matrix; (Ak+1F¯,AkF)(AkF¯,Ak+1F) (A_{k + 1}^{\bar F},A_k^F) \Rightarrow (A_k^{\bar F},A_{k + 1}^F) is called the condition of P-matrix reasoning, ((x)k+1F¯,(x)kF)((x)kF¯,(x)k+1F) ((x)_{k + 1}^{\overline F },(x)_k^F) \Rightarrow ((x)_k^{\overline F },(x)_{k + 1}^F) is called the conclusion of P-matrix reasoning. In Eq. (22), (Ak+1F¯,AkF)(AkF¯,Ak+1F) (A_{k + 1}^{\bar F},A_k^F) \Rightarrow (A_k^{\bar F},A_{k + 1}^F) is equal to Ak+1F¯AkF¯ A_{k + 1}^{\bar F} \Rightarrow A_k^{\bar F} , AkFAk+1F A_k^F \Rightarrow A_{k + 1}^F ; ((x)k+1F¯¯,(x)kF)((x)kF¯,(x)k+1F) ((x)_{k + 1}^{\overline {\bar F} },(x)_k^F) \Rightarrow ((x)_k^{\bar F},(x)_{k + 1}^F) is equal to (x)k+1F¯(x)kF¯ (x)_{k + 1}^{\overline F } \Rightarrow (x)_k^{\overline F } , (x)kF(x)k+1F (x)_k^F \Rightarrow (x)_{k + 1}^F .

By Eqs (20)–(22), we get:

Proposition 3

Internal P-matrix reasoning consists of internal P-matrix reasoning family {ifAi+1F¯AiF¯,then(x)i+1F¯(x)iF¯|iI}. \{ {{if}}\;A_{i + 1}^{\bar F} \Rightarrow A_i^{\bar F},\;{{then}}\;(x)_{i + 1}^{\bar F} \Rightarrow (x)_i^{\bar F}|i \in I\} .

Proposition 4

Outer P-matrix reasoning consists of outer P-matrix reasoning family {ifAjFAj+1F,then(x)jF(x)j+1F|jJ}. \{ {{if}}\;A_j^F \Rightarrow A_{j + 1}^F,\;{{then}}\;(x)_j^F \Rightarrow (x)_{j + 1}^F|j \in J\} .

Proposition 5

P-matrix reasoning consists of P-matrix reasoning family {if(Ak+1F¯,AkF)(AkF¯,Ak+1F),then((x)k+1F¯,(x)kF)((x)kF¯,(x)k+1F)|kK}. \{ {{if}}\;(A_{k + 1}^{\bar F},A_k^F) \Rightarrow (A_k^{\bar F},A_{k + 1}^F),\;{{then}}\;((x)_{k + 1}^{\bar F},(x)_k^F) \Rightarrow ((x)_k^{\bar F},(x)_{k + 1}^F)|k \in K\} .

Theorem 1

If there is ∇(x) ≠ ∅, then Ak+1F¯ A_{k + 1}^{\bar F} and AkF¯ A_k^{\bar F} , (x)kF¯ (x)_k^{\bar F} and ((x)kF¯(x)) ((x)_k^{\bar F} - \nabla (x)) meet internal P-matrix reasoning ifAk+1F¯AkF¯,then((x)kF¯(x))(x)kF¯ {{if}}\;A_{k + 1}^{\bar F} \Rightarrow A_k^{\bar F},\;{{then}}\;((x)_k^{\bar F} - \nabla (x)) \Rightarrow (x)_k^{\bar F} ∇(x) is the redundancy of (x)kF¯ (x)_k^{\bar F} , (x)kF¯ (x)_k^{\bar F} generates (x)k+1F¯ (x)_{k + 1}^{\bar F} , (x)k+1F¯=(x)kF¯(x) (x)_{k + 1}^{\bar F} = (x)_k^{\bar F} - \nabla (x) .

Proof

By Eqs (15) and (16) in Section 3, we get: Ak+1F¯ A_{k + 1}^{\overline F } , AkF¯ A_k^{\overline F } are generated by Xk+1F¯ X_{k + 1}^{\bar F} , XkF¯ X_k^{\bar F} ; Ak+1F¯ A_{k + 1}^{\bar F} and AkF¯ A_k^{\bar F} meet Ak+1F¯AkF¯ A_{k + 1}^{\bar F} \subseteq A_k^{\bar F} , or Ak+1F¯AkF¯ A_{k + 1}^{\bar F} \Rightarrow A_k^{\bar F} . By Eqs (1) and (9) in Section 2, we get: Xk+1F¯ X_{k + 1}^{\bar F} and XkF¯ X_k^{\bar F} meet Xk+1F¯XkF¯ X_{k + 1}^{\bar F} \subseteq X_k^{\bar F} , or Xk+1F¯XkF¯ X_{k + 1}^{\bar F} \Rightarrow X_k^{\bar F} , or exists ∇X ≠ ∅, Xk+1F¯XkF¯ X_{k + 1}^{\bar F} \Rightarrow X_k^{\bar F} is equal to (XkF¯X)XkF¯ (X_k^{\overline F } - \nabla X) \Rightarrow X_k^{\overline F } ;

Here: Xk+1F¯=XkF¯X X_{k + 1}^{\bar F} = X_k^{\bar F} - \nabla X . If Ak+1F¯AkF¯ A_{k + 1}^{\bar F} \Rightarrow A_k^{\bar F} , then (XkF¯X)XkF¯ (X_k^{\bar F} - \nabla X) \Rightarrow X_k^{\bar F} , we get Eq. (26). Under the condition Ak+1F¯AkF¯ A_{k + 1}^{\bar F} \Rightarrow A_k^{\bar F} , Xk+1F¯ X_{k + 1}^{\bar F} is generated by delete ∇X out of XkF¯ X_k^{\bar F} , ∇(x) is the redundancy of (x)kF¯ (x)_k^{\bar F} to generate (x)k+1F¯ (x)_{k + 1}^{\bar F} . Here: ∇X = ∇(x), XkF¯=(x)kF¯ X_k^{\bar F} = (x)_k^{\bar F} , Xk+1F¯=(x)k+1F¯ X_{k + 1}^{\bar F} = (x)_{k + 1}^{\bar F} .

Theorem 2

If there is Δ(x) ≠ ∅, then Ak+1F A_{k + 1}^F and AkF A_k^F , (x)kF (x)_k^F and ((x)kFΔ(x)) ((x)_k^F \cup \Delta (x)) meet outer P-matrix reasoning ifAkFAk+1F,then(x)kF((x)kFΔ(x)). {{if}}\;A_k^F \Rightarrow A_{k + 1}^F,\;{{then}}\;(x)_k^F \Rightarrow ((x)_k^F \cup \Delta (x)). Δ(x) is the supplementary of (x)kF (x)_k^F , (x)kF (x)_k^F generates (x)k+1F (x)_{k + 1}^F , (x)k+1F=((x)kFΔ(x)) (x)_{k + 1}^F = ((x)_k^F \cup \Delta (x)) .

Since the proof of Theorem 2 is similar to the proof of Theorem 1, the proof is omitted. By Theorems 1 and 2, we get Theorem 3.

Theorem 3

If there is (∇(x),Δ(x)) ≠ ∅, then (Ak+1F¯,AkF) (A_{k + 1}^{\bar F},A_k^F) and (AkF¯,Ak+1F) (A_k^{\bar F},A_{k + 1}^F) , ((x)kF¯(x),(x)kF) ((x)_k^{\bar F} - \nabla (x),(x)_k^F) and ((x)kF¯,(x)kFΔ(x)) ((x)_k^{\bar F},(x)_k^F \cup \Delta (x)) meet P-matrix reasoning if(Ak+1F¯,AkF)(AkF¯,Ak+1F),then((x)kF¯(x),(x)kF)((x)kF¯,(x)kFΔ(x)). {{if}}\;(A_{k + 1}^{\bar F},A_k^F) \Rightarrow (A_k^{\bar F},A_{k + 1}^F),\;{{then}}\;((x)_k^{\bar F} - \nabla (x),(x)_k^F) \Rightarrow ((x)_k^{\bar F},(x)_k^F \cup \Delta (x)). (∇(x),Δ(x)) is the redundancy-supplementary of ((x)kF¯,(x)kF) ((x)_k^{\bar F},(x)_k^F) , ((x)kF¯,(x)kF) ((x)_k^{\bar F},(x)_k^F) generates ((x)kF¯(x),(x)kFΔ(x)) ((x)_k^{\bar F} - \nabla (x),(x)_k^F \cup \Delta (x)) ; (x)kF¯(x)=(x)k+1F¯ (x)_k^{\bar F} - \nabla (x) = (x)_{k + 1}^{\bar F} , (x)kFΔ(x)=(x)k+1F (x)_k^F \cup \Delta (x) = (x)_{k + 1}^F .

Here: (∇(x),Δ(x)) ≠ ∅ represents ∇(x) ≠ ∅, Δ(x) ≠ ∅.

Using the concepts in Sections 2 and 3 and results in Section 4, Section 5 is given.

The information intelligent recursive mining of P-matrix reasoning and its application

Assumption: For simple and without losing generality, in this section only the application of internal P-information intelligent recursive mining under the condition of internal P-matrix reasoning is given. The application of outer P-information intelligent recursive mining under the condition of outer P-matrix reasoning and the application of P-information intelligent recursive mining under the condition of P-matrix reasoning are omitted.

Source of information (x) and its attribute set α are given: (x)={x1,x2,x3,x4,x5,x6} (x) = \{ {x_1},{x_2},{x_3},{x_4},{x_5},{x_6}\} α={α1,α2,α3,α4} \alpha = \{ {\alpha _1},{\alpha _2},{\alpha _3},{\alpha _4}\} A is the information value matrix generated by source information (x) A=(1.201.401.001.301.601.501.601.681.101.501.681.001.371.101.701.401.451.601.451.701.201.001.901.301.801.321.651.801.201.00). A = \left( {\matrix{ {1.20} & {1.40} & {1.00} & {1.30} & {1.60} & {1.50} \cr {1.60} & {1.68} & {1.10} & {1.50} & {1.68} & {1.00} \cr {1.37} & {1.10} & {1.70} & {1.40} & {1.45} & {1.60} \cr {1.45} & {1.70} & {1.20} & {1.00} & {1.90} & {1.30} \cr {1.80} & {1.32} & {1.65} & {1.80} & {1.20} & {1.00} \cr } } \right).

Here: column j of matrix A in Eq. (31) is the vector yi = (y1,j,y2,j,y3,j,y4,j,y5,j)T composed of the information values y1,j,y2,j,y3,j,y4,j,y5,j of information element xj ∈ (x), j = 1,2,3,4,5,6;

AkF¯ A_k^{\bar F} , Ak+1F¯ A_{k + 1}^{\bar F} are internal P-matrices of A AkF¯=(1.201.401.001.301.601.601.681.101.501.681.371.101.701.401.451.451.701.201.001.901.801.321.651.801.20) A_k^{\bar F} = \left( {\matrix{ {1.20} & {1.40} & {1.00} & {1.30} & {1.60} \cr {1.60} & {1.68} & {1.10} & {1.50} & {1.68} \cr {1.37} & {1.10} & {1.70} & {1.40} & {1.45} \cr {1.45} & {1.70} & {1.20} & {1.00} & {1.90} \cr {1.80} & {1.32} & {1.65} & {1.80} & {1.20} \cr } } \right) Ak+1F¯=(1.201.401.001.601.681.101.371.101.701.451.701.201.801.321.65). A_{k + 1}^{\bar F} = \left( {\matrix{ {1.20} & {1.40} & {1.00} \cr {1.60} & {1.68} & {1.10} \cr {1.37} & {1.10} & {1.70} \cr {1.45} & {1.70} & {1.20} \cr {1.80} & {1.32} & {1.65} \cr } } \right). Under the limitation of the reasoning condition Ak+1F¯AkF¯ A_{k + 1}^{\bar F} \Rightarrow A_k^{\bar F} of the internal P-matrix reasoning Eq. (20), by ifAkF¯A,then(x)kF¯(x) {\rm{if}}\;A_k^{\bar F} \Rightarrow A,\;{\rm{then}}\;(x)_k^{\bar F} \Rightarrow (x) (x)kF¯ (x)_k^{\bar F} is intelligent mined from (x), (x)kF¯(x) (x)_k^{\bar F} \subseteq (x) (x)kF¯={x1,x2,x3,x4,x5} (x)_k^{\bar F} = \{ {x_1},{x_2},{x_3},{x_4},{x_5}\} (x)k=(x)(x)kF¯={x1,x2,x3,x4,x5,x6}{x1,x2,x3,x4,x5}={x6} \nabla {(x)_k} = (x) - (x)_k^{\bar F} = \{ {x_1},{x_2},{x_3},{x_4},{x_5},{x_6}\} - \{ {x_1},{x_2},{x_3},{x_4},{x_5}\} = \{ {x_6}\} is the redundancy information of (x) to intelligent mining (x)kF¯ (x)_k^{\bar F} .

Under the condition of internal P-matrix reasoning: Ak+1F¯AkF¯ A_{k + 1}^{\bar F} \Rightarrow A_k^{\bar F} , by ifAk+1F¯AkF¯,then(x)k+1F¯(x)kF¯ {\rm{if}}\;A_{k + 1}^{\bar F} \Rightarrow A_k^{\bar F},\;{\rm{then}}\;(x)_{k + 1}^{\bar F} \Rightarrow (x)_k^{\bar F} (x)k+1F¯ (x)_{k + 1}^{\bar F} is intelligent mined from (x)kF¯ (x)_k^{\bar F} , (x)k+1F¯(x)kF¯ (x)_{k + 1}^{\bar F} \subseteq (x)_k^{\bar F} (x)k+1F¯={x1,x2,x3} (x)_{k + 1}^{\bar F} = \{ {x_1},{x_2},{x_3}\} (x)k+1=(x)kF¯(x)k+1F¯={x1,x2,x3,x4,x5}{x1,x2,x3}={x4,x5} \nabla {(x)_{k + 1}} = (x)_k^{\bar F} - (x)_{k + 1}^{\bar F} = \{ {x_1},{x_2},{x_3},{x_4},{x_5}\} - \{ {x_1},{x_2},{x_3}\} = \{ {x_4},{x_5}\} is the redundancy information of (x)kF¯ (x)_k^{\overline F } to intelligent mining (x)k+1F¯ (x)_{k + 1}^{\bar F} , k ∈ (1,2,⋯ ,n).

By using internal P-matrix reasoning family Eq. (23), (x)kF¯ (x)_k^{\bar F} , (x)k+1F¯ (x)_{k + 1}^{\bar F} are discovered from (x) by intelligent recursive mining; (x), (x)kF¯ (x)_k^{\bar F} , (x)k+1F¯ (x)_{k + 1}^{\bar F} meet IDE((x)k+1F¯,(x)kF¯,(x)) {\rm{IDE}}\;((x)_{k + 1}^{\bar F},(x)_k^{\bar F},(x)) where IDE is the identification.

Special note: Examples in this section are taken from the computer information image intelligent recognition system. Matrix A in Eq. (31) is composed of the real information image data. AkF¯ A_k^{\bar F} , Ak+1F¯ A_{k + 1}^{\bar F} in Eqs (32) and (33) are composed of two information image data, the two information images are false images of real information image. The purpose of these examples is to identify two false information images by using the information intelligent mining method generated by internal P-matrix reasoning, the identification result is given in Eq. (38), this result is confirmed in our experiments.

Discussion

The cross and penetration of P-sets and P-matrix reasoning is a new research field of information intelligent dynamic mining. In this paper, the basic theory and application of information (data) intelligent mining are given by using the intersection of P-sets and P-matrix reasoning. This paper discusses information intelligent recursive mining and mining methods, and gives its application in information image intelligent mining and recognition. Several new concepts are discussed, and a new reasoning method (intelligent method) is proposed. Ref. [12] gives the dual form of inverse P-sets of P-sets, and Ref. [13] presented inverse P-matrix, which is the dual form of P-matrix. In Refs. [14,15,16,17,18], inverse P-sets and inverse P-matrix have been applied in the dynamic discovery of unknown information and the intelligent fusion and separation of information laws. P-sets, inverse P-sets, P-matrix and inverse P-matrix provide new mathematical theories and methods for information intelligent mining, and become a new tool to study dynamic information system.

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